Solve 8y^2+17y+9

$$8 y ^ { 2 } + 17 y + 9$$

Factor

$$\left(y+1\right)\left(8y+9\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 8y^{2}+ay+by+9. To find a and b, set up a system to be solved.

$$a+b=17 ab=8\times 9=72$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 72.

$$1,72 2,36 3,24 4,18 6,12 8,9$$

Calculate the sum for each pair.

$$1+72=73 2+36=38 3+24=27 4+18=22 6+12=18 8+9=17$$

The solution is the pair that gives sum 17.

$$a=8 b=9$$

Rewrite 8y^{2}+17y+9 as \left(8y^{2}+8y\right)+\left(9y+9\right).

$$\left(8y^{2}+8y\right)+\left(9y+9\right)$$

Factor out 8y in the first and 9 in the second group.

$$8y\left(y+1\right)+9\left(y+1\right)$$

Factor out common term y+1 by using distributive property.

$$\left(y+1\right)\left(8y+9\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$8y^{2}+17y+9=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$y=\frac{-17±\sqrt{17^{2}-4\times 8\times 9}}{2\times 8}$$

Square 17.

$$y=\frac{-17±\sqrt{289-4\times 8\times 9}}{2\times 8}$$

Multiply -4 times 8.

$$y=\frac{-17±\sqrt{289-32\times 9}}{2\times 8}$$

Multiply -32 times 9.

$$y=\frac{-17±\sqrt{289-288}}{2\times 8}$$

Add 289 to -288.

$$y=\frac{-17±\sqrt{1}}{2\times 8}$$

Take the square root of 1.

$$y=\frac{-17±1}{2\times 8}$$

Multiply 2 times 8.

$$y=\frac{-17±1}{16}$$

Now solve the equation y=\frac{-17±1}{16} when ± is plus. Add -17 to 1.

$$y=-\frac{16}{16}$$

Divide -16 by 16.

$$y=-1$$

Now solve the equation y=\frac{-17±1}{16} when ± is minus. Subtract 1 from -17.

$$y=-\frac{18}{16}$$

Reduce the fraction \frac{-18}{16} to lowest terms by extracting and canceling out 2.

$$y=-\frac{9}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -1 for x_{1} and -\frac{9}{8} for x_{2}.

$$8y^{2}+17y+9=8\left(y-\left(-1\right)\right)\left(y-\left(-\frac{9}{8}\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$8y^{2}+17y+9=8\left(y+1\right)\left(y+\frac{9}{8}\right)$$

Add \frac{9}{8} to y by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$8y^{2}+17y+9=8\left(y+1\right)\times \frac{8y+9}{8}$$

Cancel out 8, the greatest common factor in 8 and 8.

$$8y^{2}+17y+9=\left(y+1\right)\left(8y+9\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 8

$$x ^ 2 +\frac{17}{8}x +\frac{9}{8} = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = -\frac{17}{8} rs = \frac{9}{8}$$

Two numbers r and s sum up to -\frac{17}{8} exactly when the average of the two numbers is \frac{1}{2}*-\frac{17}{8} = -\frac{17}{16}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -\frac{17}{16} - u s = -\frac{17}{16} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{9}{8}

$$(-\frac{17}{16} - u) (-\frac{17}{16} + u) = \frac{9}{8}$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{289}{256} - u^2 = \frac{9}{8}$$

Simplify the expression by subtracting \frac{289}{256} on both sides

$$-u^2 = \frac{9}{8}-\frac{289}{256} = -\frac{1}{256}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{1}{256} u = \pm\sqrt{\frac{1}{256}} = \pm \frac{1}{16}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =-\frac{17}{16} - \frac{1}{16} = -1.125 s = -\frac{17}{16} + \frac{1}{16} = -1$$

Evaluate

$$\left(y+1\right)\left(8y+9\right)$$